Finite unitary ring with minimal non-nilpotent group of units.

Let R be a finite unitary ring such that R = R 0 [ R ∗ ] , where R 0 is the prime ring and R ∗ is not a nilpotent group. We show that if all proper subgroups of R ∗ are nilpotent groups, then the cardinality of R is a power of 2. In addition, if (R / Jac (R)) ∗ is not a p -group, then either R ≅ M 2 (G F (2)) or R ≅ M 2 (G F (2)) ⊕ A , where M 2 (G F (2)) is the ring of 2 × 2 matrices over the finite field G F (2) and A is a direct sum of copies of the finite field G F (2). [ABSTRACT FROM AUTHOR]